In this paper we derive differential equations for evolving radial basis functions (RBFs) to solve segmentation problems. The differential equations result from applying variational calculus to energy functionals designed for image segmentation. Our methodology supports evolution of all parameters of each RBF, including its position, weight, orientation, and anisotropy, if present. Our framework is general and can be applied to numerous RBF interpolants. The resulting approach retains some of the ideal features of implicit active contours, like topological adaptivity, while requiring low storage overhead due to the sparsity of our representation, which is an unstructured list of RBFs. We present the theory behind our technique and demonstrate its usefulness for image segmentation.

Mappings between surfaces have a variety of uses, including texture transfer, multi-way morphing, and surface analysis. Given a 4D implicit function that defines a morph between two implicit surfaces, this article presents a method of calculating a mapping between the two surfaces. We create such a mapping by solving two PDEs over a tetrahedralized hypersurface that connects the two surfaces in 4D. Solving the first PDE yields a vector field that indicates how points on one surface flow to the other. Solving the second PDE propagates position labels along this vector field so that the second surface is tagged with a unique position on the first surface. One strength of this method is that it produces correspondences between surfaces even when they have different topologies. Even if the surfaces split apart or holes appear, the method still produces a mapping entirely automatically. We demonstrate the use of this approach to transfer texture between two surfaces that may have differing topologies.

We present a new method of surface reconstruction that generates smooth and seamless models from sparse, noisy, nonuniform, and low resolution range data. Data acquisition techniques from computer vision, such as stereo range images and space carving, produce 3D point sets that are imprecise and nonuniform when compared to laser or optical range scanners. Traditional reconstruction algorithms designed for dense and precise data do not produce smooth reconstructions when applied to vision-based data sets. Our method constructs a 3D implicit surface, formulated as a sum of weighted radial basis functions. We achieve three primary advantages over existing algorithms: (1) the implicit functions we construct estimate the surface well in regions where there is little data, (2) the reconstructed surface is insensitive to noise in data acquisition because we can allow the surface to approximate, rather than exactly interpolate, the data, and (3) the reconstructed surface is locally detailed, yet globally smooth, because we use radial basis functions that achieve multiple orders of smoothness.

Point sets obtained from computer vision techniques are often noisy and non-uniform. We present a new method of surface reconstruction that can handle such data sets using anisotropic basis functions. Our reconstruction algorithm draws upon the work in variational implicit surfaces for constructing smooth and seamless 3D surfaces. Implicit functions are often formulated as a sum of weighted basis functions that are radially symmetric. Using radially symmetric basis functions inherently assumes, however that the surface to be reconstructed is, everywhere, locally symmetric. Such an assumption is true only at planar regions, and hence, reconstruction using isotropic basis is insufficient to recover objects that exhibit sharp features. We preserve sharp features using anisotropic basis that allow the surface to vary locally. The reconstructed surface is sharper along edges and at corner points. We determine the direction of anisotropy at a point by performing principal component analysis of the data points in a small neighborhood. The resulting field of principle directions across the surface is smoothed through tensor filtering. We have applied the anisotropic basis functions to reconstruct surfaces from noisy synthetic 3D data and from real range data obtained from space carving.

Implicit surfaces are often created by summing a collection of radial basis functions. Researchers have begun to create implicit surfaces that exactly interpolate a given set of points by solving a simple linear system to assign weights to each basis function. Due to their ability to interpolate, these implicit surfaces are more easily controllable than traditional "blobby" implicits. There are several additional forms of control over these surfaces that make them attractive for a variety of applications. Surface normals may be directly specified at any location over the surface, and this allows the modeller to pivot the normal while still having the surface pass through the constraints. The degree of smoothness of the surface can be controlled by changing the shape of the basis functions, allowing the surface to be pinched or smooth. On a point-by-point basis the modeller may decide whether a constraint point should be exactly interpolated or approximated. Applications of these implicits include shape transformation, creating surfaces from computer vision data, creation of an implicit surface from a polygonal model, and medical surface reconstruction.